Noncommutative geometry of the Moyal plane: translation isometries, Connes' distance on coherent states, Pythagoras equality

We study the metric aspect of the Moyal plane from Connes' noncommutative geometry point of view. First, we compute Connes' spectral distance associated with the natural isometric action of R^2 on the algebra of the Moyal plane A. We show that the distance between any state of A and any of its translated is precisely the amplitude of the translation. As a consequence, we obtain the spectral distance between coherent states of the quantum harmonic oscillator as the Euclidean distance on the plane. We investigate the classical limit, showing that the set of coherent states equipped with Connes' spectral distance tends towards the Euclidean plane as the parameter of deformation goes to zero. The extension of these results to the action of the symplectic group is also discussed, with particular emphasize on the orbits of coherent states under rotations. Second, we compute the spectral distance in the double Moyal plane, intended as the product of (the minimal unitization of) A by C^2. We show that on the set of states obtained by translation of an arbitrary state of A, this distance is given by Pythagoras theorem. On the way, we prove some Pythagoras inequalities for the product of arbitrary unital & non-degenerate spectral triples. Applied to the Doplicher-Fredenhagen-Roberts model of quantum spacetime [DFR], these two theorems show that Connes' spectral distance and the DFR quantum length coincide on the set of states of optimal localization. Some of the results of this paper can be thought as a continuation of arXiv:0912.0906, as well as a companion to arXiv:1106.0261.